Question 1
Use integration by substitution to solve the integral below.
\[ \int 4e^{-7x} \, dx \]

Answer

u=-7x
du=-7dx
dx=-(1/7)du

\[ \begin{align*} \int 4e^{-7x} \, dx &= \int 4e^u \frac{-1}{7}du \\ &= -\frac{4}{7} \int e^u \, du \\ &= -\frac{4}{7} e^u + C \quad \\ &= -\frac{4}{7} e^{-7x} + C \quad \ \end{align*} \]

Question 2 Biologists are treating a pond contaminated with bacteria. The level of contamination is changing at a rate of dN/dt = -3150 / t4 - 220 bacteria per cubic centimeter per day, where t is the number of days since treatment began. Find a function N( t ) to estimate the level of contamination if the level after 1 day was 6530 bacteria per cubic centimeter.

library(pracma)
dN_dt = function(t){
  -3150 / t^4 - 220
  }

n_t = function(t) {
    int = integral(dN_dt, 1, t)
    n_1 = 6530
    n_t = int + n_1
    return(n_t)
}

n_t(8)
## [1] 3942.051

Question 3 Find the total area of the red rectangles in the figure below, where the equation of the line is f ( x ) = 2x - 9.

Area of rectangle = length * width
The length of each red rectangle is one and the width of each red rectangle is 1.
There are 16 rectangles: 16 (1*1)=16.
Total area of the red rectangles =16.

Question 4 Find the area of the region bounded by the graphs of the given equations. y = x2 - 2x - 2, y = x + 2

\[ x^2 - 2x - 2 = x + 2 \] \[ x^2 - 2x - 2 - x - 2 = 0 \] \[ x^2 - 3x - 4 = 0 \] \[ x^2 - 3x - 4 = (x - 4)(x + 1) = 0 \] \[x = 4, x = -1\]

#Find area in-build function
y_1 = function(x) {x + 2}
y_2 = function(x) {x^2 -2*x -2}

#Find the difference between areas under the curve
area_y_1 <- integrate(y_1, -1, 4)
area_y_2 <- integrate(y_2, -1, 4)

area_y_1$value
## [1] 17.5
area_y_2$value
## [1] -3.333333
answer<-area_y_1$value -area_y_2$value
answer
## [1] 20.83333

Question 5 A beauty supply store expects to sell 110 flat irons during the next year. It costs $3.75 to store one flat iron for one year. There is a fixed cost of $8.25 for each order. Find the lot size and the number of orders per year that will minimize inventory costs.

expected= 110
order_cost=8.25
storage_cost=3.75

order_count=sqrt((2*expected*order_cost)/storage_cost)
order_number=expected/order_count

cat("Lot size that will minimize inventory costs is", order_count, "\n")
## Lot size that will minimize inventory costs is 22
cat("Number of orders that will minimize inventory costs is", order_number, "\n")
## Number of orders that will minimize inventory costs is 5